reserve a, b, c, d, x, y, z for Complex;
reserve r for Real;

theorem Th52:
  for a being Complex st a <> 0 ex i being Integer st Arg(
  Rotate(a,r)) = 2*PI*i+(r+Arg(a))
proof
  let a be Complex;
A1: |.a.| = |.Rotate(a,r).| by Th51;
  assume a <> 0;
  then
A2: Rotate(a, r) <> 0c by Th50;
  take -[\ (r+Arg a)/(2*PI) /];
  consider AR being Real such that
A3: AR = 2*PI*-[\ (r+Arg a)/(2*PI) /]+(r+Arg a) and
A4: 0 <= AR & AR < 2*PI by Th1,COMPTRIG:5;
  cos (r+Arg a) = cos AR & sin (r+Arg a) = sin AR by A3,Th8,Th9;
  hence thesis by A2,A1,A3,A4,COMPTRIG:def 1;
end;
